Simple Linear Regression

Our goal is predict the number of fawn given the number of adults so we can forecast Antelope population. The intuition is that the number of adults is a good indicator of the number of fawns to be born.

Simple Linear Regression (vs. correlation)

Determines the association of two variables, but no indication of their numerical depedency

Interpreting the Output - Coefficients

Coefficients: Represent the intercept and slope terms in the linear model.

Intercept: The expected value of y when we all other variables are held constant (Predict number of fawns if there were 0 adults is -1.6791364)

Slope: The effect the independent variable has on the outcome.(For each one unit increase in the number of adult the number of fawns increases (or decreases if the estimate is negative) by 0.4975309)

## (Intercept) adult
## -1.6791364 0.4975309

Interpreting the Output - Coefficients

\(\mathbf{p-value}\): indicates the extent to which a coefficient is statistically significant.

Interpret \(p-value\) as the probability that, given a chance model, results as extreme as the observed results could occur

Lower p-values are better and the cutoff for significance is normally \(<=\) 0.05, but may vary depending on field of study.

Interpreting the Output - Model Performance

Two approaches to assess the overlall model:

\(\mathbf{p-value}\): indicates the extent to which a coefficient is statistically significant. - We can consider a linear model to be statistically significant only when both these p-Values are less that the pre-determined statistical significance level

\(\mathbf{R^2}\) (coefficient of determination): ranges from 0 to 1 and measures the proportion of variation in data accounted for in the model. “How well the model fits the data”.

In fawn model \(R^2\) = 0.8813404 and adjusted \(R^2\) = 0.8615638

Reporting/interpreting the results of simple linear regression

An example of explaining the relationship between fawn and adults to be written in text:

“In modeling the fawns population it was found that the number adults (\(\beta\) = 0.5, p < .001) was a significant predictor. The overall model fit was \(R^2\) = 0.88.”